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Gauss Elimination: Mathematical Derivation (Part 2)

Figure 1: Visualizing the matrix transformation process. Deep Dive into the Algorithm In the previous post (Part 1) , we introduced the basic concept and Excel VBA code for the Gauss Elimination Method . In this post, we will look under the hood at the mathematical derivation that makes this algorithm work. Understanding these steps is critical for engineers who want to write their own solvers or understand why simulation software sometimes fails (e.g., division by zero errors). Search for Numerical Methods with MATLAB Advertisement Phase 1: Forward Elimination Why do we call it "Elimination"? Because our goal is to systematically remove variables from equations until we are left with a solvable state. Let's consider the general form of a system of linear equations: Step 1: Normalization The algorithm starts by normalizing the first equation. We divide the entire Equation (1) by...

Solving System of Equations using Gauss Elimination Method (Part 1)

The Backbone of Engineering Simulation In mechanical engineering, solving a system of linear equations is perhaps the most fundamental calculation we perform. Whether you are running a Finite Element Analysis (FEA) to check stress concentrations or a Computational Fluid Dynamics (CFD) simulation, the computer is ultimately solving a massive system of equations in the form of [A]{x} = {B} . The Gauss Elimination Method is a classic algorithm used to solve these systems. Unlike Cramer's Rule, which is inefficient for large matrices, Gauss Elimination scales well for complex engineering problems. Search for Numerical Methods for Engineers Books Advertisement Step 1: Setting up the Matrix To solve a system of equations in Excel, we first organize our coefficients into a matrix format. As shown in the figure below, the coefficients of the variables (x, y, z) form the [A] Matrix , while the constants on the right ...

5th-Degree Polynomial Cam Curve Derivation (Part 2)

Figure 1: Deriving the "secret sauce" of smooth motion control requires calculus. In [ Polynomial Cam Function (Introduction) - Part 1 ], we discussed the fundamental law of cam design: continuity of acceleration. Advertisement In this post, we are going to derive the exact equations for the Fifth-Degree Polynomial Cam Function . This is the mathematical foundation used in Motion Control Algorithms for high-end servo drives to ensure smooth, jerk-free movement. The General Equations We start with the general polynomial equation. To make the math handleable, we normalize the input angle as a ratio (x = θ / β) , where x goes from 0 to 1. s = C 0 + C 1 x + C 2 x 2 + C 3 x 3 + C 4 x 4 + C 5 x 5 Where: s = Displacement (mm) x = Ratio of cam angle (θ / β) β = Total angle in sector (rad) To find Velocity (v) and Acceleration (a) , we differentiate with respect to the angle. (Note: The ch...

Timing Diagram (Part 2 - Maximum acceleration calculation)

In the previous post [ Timing Diagram (Part 1 - No Overlap Movement) ], we determined that without overlap, our die must travel 50mm within a tight cam angle of just 55 degrees. Advertisement Now, we must ask: What is the physical cost of this rapid movement? To answer this, we calculate the Maximum Acceleration . In Machine Dynamics , acceleration is directly proportional to Force (F = m × a). High acceleration means high inertial forces, which lead to severe wear, vibration, and the need for expensive oversized servo motors . Step 1: The Time Calculation First, we need to convert our "Cam Angle" into actual "Time" in seconds. Let: N = Machine Speed (pieces per hour) B m = Indexing Angle (degrees) Cycle time (sec) = 3600 / N Indexing time t m = (B m / 360) × Cycle time Indexing time t m = (B m / 360) × (3600 / N) t m (sec) = (10 × B m ) / N Step 2: Cycloid Cam Profile Equations The...

FEA Solution Phase: Stiffness Matrices, Solvers, and Convergence

The following four-article series serves as a comprehensive introduction to the analysis discipline known as the finite element method (FEM). Originally based on works by engineering consultant Steve Roensch, this guide has been updated to cover modern solver algorithms and hardware acceleration. Third in a four-part series. Previous: Part 2: Pre-processing & Meshing Advertisement The Solution Phase: Solving the Matrix While the pre-processing and post-processing phases are interactive and human-intensive, the solution phase is a computational batch process. It is the "black box" where the computer does the heavy lifting. This phase is the primary bottleneck for complex models. For professional analysis, we recommend using dedicated Mobile Workstations or high-performance desktops equipped with ample RAM (64GB+) and, increasingly, GPU acceleration (CUDA) to speed up matrix operations. The Governing Equation: [K]{d} = {...

Numerical Methods - The Newton-Raphson Method (Part 2)

In the previous post , we talked about several root finding techniques. In this post, we're going to see how we can use Microsoft Excel VBA to find the roots using the Newton-Raphson Method. Advertisement Figure 1: The Newton-Raphson method rapidly converges on a solution using the function's derivative. The Logic As we know, the Newton-Raphson Method is the most widely used of all root-locating formulas. It uses the slope ( first derivative ) of the function to find the root. That means, in the VBA code, we have to calculate the first derivative of the function. We already discussed how to find the first derivative using numerical methods , and we will incorporate that logic here. Figure 2: Geometric interpretation: The tangent line points to the next approximate root. The Iterative Formula: The method uses the Taylor series to approximate the next position (x N ) based on the curre...

Calculate First Derivatives with Numerical Method Using Excel VBA

In the previous post, Numerical Methods - First derivative using Excel formula , we learned how to calculate the first derivative of functions using standard formulas in the spreadsheet grid. While effective, that method can be cumbersome to set up for repeated use. In this post, let's see how we can simplify the process by creating a reusable custom function using Excel VBA (Visual Basic for Applications) . Instead of downloading a pre-made file, follow the steps below to build this powerful tool yourself. Advertisement The VBA Code Solution The following code implements the Five-Point Stencil method for high precision. It defines the mathematical function f(x) and a derivative function fDeriv(x) . Step 1: Open Excel and press Alt+F11 to open the VBA Editor. Step 2: Go to Insert > Module . Step 3: Copy and paste the code below into the module window. ' ================================================ ' Created ...

Numerical Methods - The Newton-Raphson Method (Part 1)

Machine designers frequently deal with complex equations in their design projects . While some roots can be found directly, many algebraic and transcendental equations require numerical approximation. Advertisement For example, the classical equation f(x) = e -x - x cannot be solved analytically. In these cases, engineers rely on robust Root Finding Algorithms . Figure 1: Numerical methods approximate the point where the function crosses zero. These algorithms generally fall into two categories: Bracketing Methods and Open Methods . 1. Bracketing Methods Bracketing methods require two initial guesses that must "bracket" the root (one positive, one negative relative to the root). They are reliable but often slower. The Bisection Method: An incremental search based on sign changes. It repeatedly cuts the interval in half. Also known as binary chopping or Bolzano's method . The False-Position Me...

Stress Analysis Explained: Mohr's Circle, Tresca, and Von Mises

The German engineer Otto Mohr (1835-1918) developed a useful pictorial interpretation of the equations for finding principal stresses and maximum shearing stress at a point in a stressed member. Advertisement This method, commonly called Mohr's Circle , involves constructing a circle where the coordinates of each point represent the normal and shearing stresses on a specific plane. The angular position of the radius gives the orientation of that plane. Understanding the Plot Figure 1: The geometric relationship between Normal Stress (σ) and Shear Stress (τ). Coordinate Rules: Normal Stresses (σ): Plotted on the horizontal axis. Tensile (+) is right; Compressive (-) is left. Shearing Stresses (τ): Plotted on the vertical axis. Clockwise rotation is above the axis; Counter-clockwise is below. The results obtained from Mohr's circle are identical to the equations derived from the free-body diagram. ...